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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Surfaces expanding by the power of the Gauss curvature flow
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by Qi-Rui Li PDF
Proc. Amer. Math. Soc. 138 (2010), 4089-4102 Request permission

Abstract:

In this paper, we describe the flow of 2-surfaces in $\mathbb {R}^{3}$ for some negative power of the Gauss curvature. We show that strictly convex surfaces expanding with normal velocity $K^{-\alpha }$, when $\frac {1}{2}<\alpha \leq 1$, converge to infinity in finite time. After appropriate rescaling, they converge to spheres. In the 2-dimensional case, our results close an apparent gap in the powers considered by previous authors, that is, for $\alpha \in (0,\frac {1}{2}]$ by Urbas and Huisken and for $\alpha =1$ by Schnürer.
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Additional Information
  • Qi-Rui Li
  • Affiliation: Department of Mathematics, Zhejiang University, Hangzhou 310027, People’s Republic of China
  • Email: 85lqr@163.com
  • Received by editor(s): June 2, 2009
  • Received by editor(s) in revised form: October 9, 2009, November 26, 2009, December 19, 2009, December 31, 2009, and February 6, 2010
  • Published electronically: June 16, 2010
  • Communicated by: Richard A. Wentworth
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 138 (2010), 4089-4102
  • MSC (2010): Primary 53C44, 35K55
  • DOI: https://doi.org/10.1090/S0002-9939-2010-10431-8
  • MathSciNet review: 2679630